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Properties

Label 2-338-13.12-c3-0-28
Degree $2$
Conductor $338$
Sign $-0.969 + 0.246i$
Analytic cond. $19.9426$
Root an. cond. $4.46571$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 8.74·3-s − 4·4-s − 14.1i·5-s − 17.4i·6-s − 28.6i·7-s − 8i·8-s + 49.4·9-s + 28.3·10-s − 9.49i·11-s + 34.9·12-s + 57.3·14-s + 123. i·15-s + 16·16-s − 30.6·17-s + 98.8i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.68·3-s − 0.5·4-s − 1.26i·5-s − 1.18i·6-s − 1.54i·7-s − 0.353i·8-s + 1.82·9-s + 0.896·10-s − 0.260i·11-s + 0.841·12-s + 1.09·14-s + 2.13i·15-s + 0.250·16-s − 0.436·17-s + 1.29i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ -0.969 + 0.246i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5456851244\)
\(L(\frac12)\) \(\approx\) \(0.5456851244\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
13 \( 1 \)
good3 \( 1 + 8.74T + 27T^{2} \)
5 \( 1 + 14.1iT - 125T^{2} \)
7 \( 1 + 28.6iT - 343T^{2} \)
11 \( 1 + 9.49iT - 1.33e3T^{2} \)
17 \( 1 + 30.6T + 4.91e3T^{2} \)
19 \( 1 + 153. iT - 6.85e3T^{2} \)
23 \( 1 - 36.0T + 1.21e4T^{2} \)
29 \( 1 + 49.2T + 2.43e4T^{2} \)
31 \( 1 + 166. iT - 2.97e4T^{2} \)
37 \( 1 + 23.8iT - 5.06e4T^{2} \)
41 \( 1 + 125. iT - 6.89e4T^{2} \)
43 \( 1 - 434.T + 7.95e4T^{2} \)
47 \( 1 + 186. iT - 1.03e5T^{2} \)
53 \( 1 + 400.T + 1.48e5T^{2} \)
59 \( 1 + 408. iT - 2.05e5T^{2} \)
61 \( 1 + 603.T + 2.26e5T^{2} \)
67 \( 1 - 287. iT - 3.00e5T^{2} \)
71 \( 1 - 961. iT - 3.57e5T^{2} \)
73 \( 1 - 963. iT - 3.89e5T^{2} \)
79 \( 1 + 1.04e3T + 4.93e5T^{2} \)
83 \( 1 + 313. iT - 5.71e5T^{2} \)
89 \( 1 - 675. iT - 7.04e5T^{2} \)
97 \( 1 - 272. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89316099632382696977923570539, −9.799612390223754790493087026215, −8.807943823404423536844026533019, −7.45777265442682667908710421938, −6.78070033283171784965039754040, −5.69566850136974958294628499577, −4.75934981103368262033062071730, −4.23020060065756259639287806827, −0.947216223449310020092715482101, −0.32760209980606574047374358591, 1.75828585826602940925284455950, 3.12273278575924337326988128205, 4.67535049577467621875267928653, 5.82494172787040893585116761241, 6.28632692633012346121557681730, 7.58308146405157944733447923447, 9.066819091042602007458034366574, 10.15184568859025939554749098314, 10.78523234423924336653178581514, 11.49077109809930448591793382012

Graph of the $Z$-function along the critical line